Suppression of decay widths in singly heavy baryons induced by the $U_A (1)$ anomaly

We study strong and radiative decays of excited singly heavy baryons (SHBs) using an effective chiral Lagrangian based on the diquark picture proposed in Ref. [1]. The effective Lagrangian contains a $U_A (1)$ anomaly term, which induces an inverse mass ordering between strange and non-strange SHBs with spin-parity $1/2^-$. We find that the effect of the $U_A (1)$ anomaly combined with flavor-symmetry breaking modifies the Goldberger-Treiman relation for the mass difference between the ground state $\Lambda_Q (1/2^+)$ and its chiral partner $\Lambda_Q (1/2^-)$, and $\Lambda_Q (1/2^-) \Lambda_Q (1/2^+) \eta$ coupling, which results in suppression of the decay width of $\Lambda_Q (1/2^-) \to \Lambda_Q (1/2^+) \eta$. We also investigate the other various decays such as $\Lambda_Q (1/2^-) \to \Sigma_Q (1/2^+, \, 3/2^+) \pi \pi$, $\Lambda_Q (1/2^-) \to \Sigma_Q (1/2^+) \pi$, $\Lambda_Q (1/2^-) \to \Sigma_Q (1/2^+, \, 3/2^+) \gamma$, and $\Lambda_Q (1/2^-) \to \Lambda_Q (1/2^+) \pi^0$ for wide range of mass of $\Lambda_Q (1/2^-)$.


I. INTRODUCTION
Spontaneous chiral symmetry breaking and the U A (1) anomaly are the essential properties of quantum chromodynamics (QCD). Since colored quarks and gluons are not directly observed at the low-energy scale in QCD, verification of these properties in hadronic phenomena provides precious clues to understand the symmetry properties of QCD. Chiral partner structure of hadron spectra and the heavy η mass spectrum are known as the examples of such phenomena.
It is also important to interpret hadronic phenomena based on colored constitutions such as diquarks. The diquark is the simplest colored cluster, that is known to play important roles in structures of baryons and exotic multi-quark hadrons, and color superconducting phase.
Singly heavy baryons (SHBs) are considered and studied as the bound states of a diqaurk and a heavy quark (c or b quark). Recently, diquarks made of light quarks are studied from the chiral-symmetry viewpoints and a chiral effective theory for scalar/pseudo-scalar diquarks was proposed [1]. The proposed Lagrangian contains a term representing U A (1) anomaly effect. It is found that the term induces the inverse mass ordering between strange and non-strange SHBs with spin parity 1/2 − .
In this paper, we focus on investigation into decay widths of SHBs with spin-parity 1/2 − based on the model given in Ref.

II. MASSES AND GOLDBERGER-TREIMAN RELATIONS
In Ref.
[1], a chiral effective Lagrangian of scalar and pseudo-scalar diquarks based on chiral SU (3) R × SU (3) L symmetry is proposed. Each diquark with a heavy quark Q (c/b) makes an SHB as a bound state which belongs to the flavor3 representation (Λ c/b or Ξ c/b ). In this paper, we express those SHBs by linear representations: S R,i (i = 1, 2, 3) belongs to (3,1)  Lagrangian of the SHBs in the chiral limit is given as where v µ is a velocity of SHBs, M B0 , M B1 , and M B2 are model parameters, f = 92.4 MeV is the pion decay constant, the indices i, j, k, l, m, n = 1, 2, 3 are for either SU (3) R or SU (3) L , and summations over repeated indices are understood. Σ ij denotes the effective field for light scalar and pseudo-scalar mesons belonging to the chiral (3,3) representation. These fields transform as The Lagrangian is invariant under these chiral transformations. In addition, the kinetic, M B0 -, and M B2 -terms are also invariant under the following U A (1) transformations: In contrast, the M B1 -term is not invariant under these transformations, reflecting the U A (1) anomaly. The chiral symmetry is spontaneously broken by the vacuum expectation values of Σ ij field as Σ ij = f δ ij . Then, the M B1 -and M B2 -terms give contributions to the mass splitting between parity eigenstates of SHBs defined as In this paper, we follow the prescription adopted in Ref.
[1], in which the explicit breaking of flavor symmetry is introduced by the replacement, with A ∼ 5/3 being the parameter of flavor breaking. The vacuum expectation value ofΣ is given by Then, the masses of the SHBs are expressed as where M ± 1,2 denote the masses of Ξ Q (1/2 + ) and Ξ Q (1/2 − ), and M ± 3 the masses of Λ Q (1/2 + ) and Λ Q (1/2 − ). From Eqs. (10) and (11), we obtain mass differences between chiral partners as We require M + 1,2 > M + 3 consistently with the experimental values of the masses of the ground-state SHBs. The inverse mass ordering between strange and non-strange SHBs proposed in Ref.
Next, we study the couplings among chiral partners and a pseudo Nambu-Goldstone (pNG) boson. For this purpose, we introduce light scalar mesons σ ij and pseudoscalar mesons π ij as with (15) The M B1 -and M B2 -terms of the Lagrangian (1) provide interactions of SHBs with light mesons. In the chiral limit (A = 1), we obtain the relation between the coupling constant for the interaction of SHBs with a pNG boson g πSP and the mass difference of SHBs ∆M as This is often called the extended GT relation. We focus on studying the coupling of Λ Q (1/2 − )Λ Q (1/2 + )η in this section, and we use g for the coupling constant below. When the flavor-symmetry breaking is included by A > 1, the coupling constant is obtained as From inverse mass ordering, we obtain ∆M 1,2 < ∆M 3 as we showed above. Then, we see that which indicates that the value of the coupling constant is smaller than the one expected from the GT relation. In order to see that this is caused by the effect of anomaly, we drop M B1 in Eqs. (13) and (17). Then, we obtain the coupling constant as which is the one expected from the GT relation in Eq. (16). Therefore, we conclude that the U A (1) anomaly suppresses the value of coupling constant. This suppression is expected to be seen in the decay width of In this section, we numerically study how the effect of anomaly suppresses the decay of where η 8 is a member of the octet of SU (3) flavor symmetry, and η 1 belongs to the flavor singlet. The realistic η is known as a mixing state of η 8 and η 1 . As shown in the previous section, although η 8 is a pNG boson associated with the chiral SU (3) R × SU (3) L symmetry breaking, its coupling constant to Λ c (1/2 − ) and Λ c (1/2 + ) given in Eq. (17) is smaller than the naive expectation of the GT relation in Eq. (16). On the other hand, the coupling constant of η 1 is read from Lagrangian (1) as This is also different from the GT relation, since η 1 is no longer a pNG boson when the effect of U A (1) anomaly is included. In the following, we first see that the coupling constant in Eq. (17) Fig. 1. The thin-solid green and thin-dotted orange curves are plotted without η-η mixing. The thin-solid green curve is drawn by using the coupling constant in Eq. (17) which includes the effect of anomaly, while the thin-dotted orange curve is by the one in Eq. (19) without anomaly. One can easily see that the thin-solid green curve is much suppressed compared with the thin-dotted orange curve.
Typical forms of interaction Lagrangians are given in Appendix A. The coupling constant of Λ Q (1/2 − ) → Σ ( * ) Q ππ is determined as k = 1 by the ρ meson dominance and the coupling universality [8][9][10]. On the other hand, the coupling constants of Λ c (1/2 − ) → Σ c π and Λ c (1/2 − ) → Σ ( * ) c γ are unknown. We therefore leave κ and r as free parameters, while for the plots and the total decay widths in Tables I and II, we use κ = r = 1 for the reference values.
We show the dependence of estimated widths on the mass of Λ c (1/2 − ) in Fig. 3. We also show the estimated widths of Λ b (1/2 − ) in Fig. 4.
We see that decay widths of Λ Q γ, and Λ Q (1/2 − ) → Λ Q (1/2 + )π 0 in the charm sector are almost the same as those in the bottom sector. On the other hand, the decay width of Λ b (1/2 − ) → Σ b π is much suppressed compared with that of Λ c (1/2 − ) → Σ c π since the heavy-quark symmetry is well satisfied in the bottom sector. Figure 3 shows that, below the threshold of Λ c (1/2 + )η indicated by the vertical thick-dash-dotted magenta line, the dominant decay mode is Λ c (1/2 − ) → Σ c π shown by the thick-dashed gray curve which violates the heavyquark symmetry. Figure 4 shows that, below the threshold, the dominant decay mode is Λ b (1/2 − ) → Σ ( * ) b ππ indicated by the thick-solid orange curve. When the mass of Λ b (1/2 − ) is a little above the threshold of decay is strongly suppressed by the heavy-quark symmetry in the bottom sector, the width is comparable to that of the radiative decay.

V. A SUMMARY AND DISCUSSIONS
In this paper, we study strong and radiative decays of excited SHBs using a chiral effective Lagrangian based on the diquark picture proposed in Ref.
[1]. We show predictions of widths for typical choices of the mass of Λ Q (1/2 − ) in Tables I and II. Our prediction on the width of Λ Q (1/2 − ) → Λ Q (1/2 + )η is strongly suppressed by the effect of anomaly compared with the prediction without anomaly. In general, the large width of the chiral partner state is an obstacle against its observation. The suppression of the width may enable us to observe the state easily. Tables I and II also show predictions on the other decay modes which are useful for experimental observation of chiral partner when the mass of Λ Q (1/2 − ) locates below the threshold of Λ Q (1/2 + )η.
In Tables I and II, we see that the radiative decay is about 10 % of the total width in the charm sector, and it is about 50 % in the bottom sector. In the chiral partner structure using (3,3) representations for the diquark with J P = 1 ± , the chiral partners to Σ Q (J P = 1/2 + ) and Σ * Q (3/2 + ) are Λ Q1 (1/2 − ) and Λ * Q1 (3/2 − ), respectively as in Eq. (A1) in Appendix A. In such a case, Q γ decays share a common coupling constant with Λ ( * ) Q1 → Λ Q (1/2 + )γ decays. Once the chiral partner Λ ( * ) Q1 is identified with some physical particles such as in Refs. [4,5], we can check the chiral partner by seeing the radiative decays of those particles.
In Table I, we can see that the width of Λ c (1/2 − ) → Λ c (1/2 + )π 0 decay is negligibly small compared with the total width. On the other hand, in Table II, Λ b (1/2 − ) → Λ b (1/2 + )π 0 decay provides non-negligible contribution to the total decay width because Λ b (1/2 − ) → Σ b π decay is much suppressed by the heavy-quark symmetry. Then, we may check the effect of anomaly through this decay when the mass of Λ b (1/2 − ) locates below the threshold of Λ b (1/2 + )η. I. Decay widths of Λc(1/2 − ) without and with the effect of anomaly. Units of masses and widths are in MeV. To estimate the total widths, we use k = 1, κ = 1, and r = 1.